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Full text of "Analytical Mechanics"

108 ANALYTICAL MECHANICS the mass by the component of the acceleration along the same direction. 98. Equilibrium as a Special Case of Motion. — When the right-hand member of the force equation vanishes, that is, when the acceleration is nil, the resultant force vanishes. But this is the condition of the equilibrium of a particle, therefore equilibrium is a case of motion in which acceleration is zero. For the equilibrium of a particle it is necessary that the resultant force vanish, but this condition is not sufficient because while the acceleration vanishes when F = 0, the velocity may have any constant value. In other words a particle may be in motion even when the resultant of the forces which act upon it vanishes. Therefore in order that a particle stay at rest not only must the resultant of the forces vanish but it must be at rest at the time of application of these forces. 99. Dimensions of Force. — In discussing the equilibrium of bodies we only compared forces because it was all that was necessary; besides we had no means of expressing forces in terms of other physical magnitudes. But now the force equation enables us to express forces in terms of the three fundamental magnitudes and thus to connect them with other physical quantities. If we substitute the dimensions of mass and acceleration in the force equation we obtain the following dimensional formula for force : 100. Units of Force. — The C.G.S. unit of force is the dyne. It is a force which gives a body of one grain mass a unit acceleration. This is denoted symbolically by the following formula: The British unit of force is the pound, which we have already defined (p. 76) as the weight, in London, of a body